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\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
\title{复变函数练习4.4 - 解析函数的零点的孤立性与唯一性 }
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\date{2024 年 5 月 20 日}
%\date{March 9, 2021}

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\begin{document}

\maketitle

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\begin{enumerate}

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\item  %Problem 01
设 $f(z)$ 在区域 $D$ 内解析，不恒为零，设 $a\in D$. 证明：$f(z)$ 以 $a$ 为 $m$ 阶零点的充分必要条件是 $f(z)=(z-a)^m\varphi(z)$, 其中 $\varphi(z)$ 在点 $a$ 的邻域 $|z-a|<R$ 内解析，且 $\varphi(a)\neq 0$. 

\vspace{0.2cm}

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\item  %Problem 02
求 $f(z)=z-\sin(z)$ 在原点的零点阶数。

\vspace{0.2cm}

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\item  %Problem 03
求 $f(z)=\sin(z)-1$ 的全部零点和阶数。

\vspace{0.2cm}

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\item  %Problem 04
（孤立性定理）设解析函数 $f(z)$ 以 $a$ 为零点，且在圆盘 $|z-a|<R$ 内不恒为零。则存在 $0<r<R$, 使得 $f(z)$ 在 $0<|z-a|<r$ 中没有零点。

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\item  %Problem 05
（唯一性定理）设函数 $f_1(z)$ 和 $f_2(z)$ 在区域 $D$ 内解析，设 $D$ 内存在无穷点列 $\{z_n\}$ 收敛于 $a\in D$, 且有
$f_1(z_n)=f_2(z_n)\, (n\ge 1)$. 证明 $f_1(z)=f_2(z)$ 对 $z\in D$ 都成立。

\vspace{0.2cm}

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\item  %Problem 06
设函数 $f_1(z)$ 和 $f_2(z)$ 在区域 $D$ 内解析。设乘积函数 $f_1(z)f_2(z)$ 在 $D$ 内恒等于零。证明 $f_1(z)$ 或 $f_2(z)$ 在 $D$ 内恒等于零。

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\item  %Problem 07
应用唯一性定理，在区域 $|z|<1$ 内，将 $\mathrm{Ln}(1+z)$ 的主值支展开成 $z$ 的幂级数。

\vspace{0.2cm}

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\item  %Problem 08
（最大模原理）设 $f(z)$ 在区域 $D$ 内解析，且不为常数。则 $|f(z)|$ 在 $D$ 内不能取到最大值。

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\item  %Problem 09
用最大模原理证明。设函数 $f(z)$ 在闭圆 $|z|\le R$ 上解析，设存在 $a>0$ 使得 $|f(0)|<a$, 且当 $|z|=R$ 时有 $|f(z)|>a$. 则 $f(z)$ 在 $|z|<R$ 至少有一个零点。

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\item  %Problem 10
求下述函数在零点 $z=0$ 的阶数，
$$
(1) f(z)=z^2(\exp(z^2)-1); \hspace{1cm}
(2) f(z)=6\sin(z^3)+z^3(z^6-6). 
$$


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\end{enumerate}


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\end{document}

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